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BIOMECH - Bending and Beam theory (measuring bending resistance (measuring…
BIOMECH - Bending and Beam theory
what happens when a beam is bent?
the largest forces set up are in the longitudinal direction
tension is set up on the convex side
compression is set up on the concave side
the size of stresses rises linearly either side of the neutral axis (or center of area)
consequences of the mechanics of beams
because of the squared term in the equation of I, the bending rigidity and strengths of a beam are strongly dependent on its...
...radius
Rigidity proportional R^4
strength proportional R³
:. its more efficient to resist bending with a single wide beam than many narrow ones
this is why trees and other plants have just a single stem and why horses and other ungulates have lost their outer toes
mass proportional R²
...shape. beams with more material further away from the center are better
to resist bending in one plane it's best to be as deep in that plane as possible and expanded top and bottom
material design within beams
since the largest forces set up are in the longitudinal direction, in many beams, the material is also aligned longitudinally
in trees, 90% of the cells are orientated longitudinally
measuring bending resistance
the easiest way to find the bending rigidity and bending strength of a beam is to carry out a 3 point bending test
deflection at center y=FL³/48EI
:. rigidity EI=dF/dy(L³/48)
strength = FmaxL/4
measuring material properties from bending test
second moment of area can be readily calculated using calculus for beams of regular cross sections
:. the youngs modulus and the breaking stress of the material can be readily calculated
the resistance to bending
resistance is largely due to longitudinal stiffness of the material but material further away from the neutral axis is more effective
1) it is stretched or compressed more
2) it acts a greater distance from the neutral axis
:. bending rigidity a∫zy²dy
the formulae for bending resistance
bending rigidity = EI
E= youngs modulus
I= 2nd movement of area = a∫zy²dy
then a simple formula links to curvature (c) to applied movement (M)
c=M/EI
and the maximal longitudinal stress (σmax) in the beam (at its outer edge)
σmax= Mymax/I
beams to resist bending in...
...one plane
man has developed 'I' beam girders (RSJ), while lateral tree roots have a figure of 8 shape
...all planes
to resist bending in all directions a hollow tube is best
both man and nature have exploited this
function of material at the neutral axis
from a stiffness point of view it would be better to do without material at the neutral axis...
...however if the center material wasn't there the outer edges would act as separate beams and less effective
in particular when bent the inner edges would bulge outwards and the outer edge move inwards
the central bit keeps the edges equidistant
consequences for tubular beams
no material in the middle to keep edges separate :. the section needs to be ovalise
the second movement of area gets less and the stresses get higher, finally the walls collapse inwards. this result is local buckling
preventing local buckling
many design solution to prevent buckling and so maintain the advantages of tubular construction
bulkheads actually act in tension, just like the spokes of bicycle wheels
stringers increase bending resistance of wall sections while bulkheads maintain a constant cross section
stringers in nature
cecropia
hedgehog spine
horsetail stem
bannana petiole
foams and fillers in nature
many biological tubes are held apart by lightweight foams or struts .
e.g. bird bones and rush stems
low cost fillers in nature
even when no lightweight fillings possible, organisms may make use of cheap filling such as water in the center
herbaceous plant stems have their centre filled with cheap parenchyma tissue, which is mostly water reinforcing fibres and collenchyma are around the edge
problems with anisotropic materials
leterial stresses can be a problem when the material making up the beam is anisotropic
in wood 80-95% of the cells are orientated longitudinally
5-20% are arranged radially
none are arranged tangentially
StrengthL>>StrengthR>>StrengthT
40MPa>>8MPa>>6MPa
transverse stresses are v. important
Case 1: curved "Hazard beam" branches
forces tending to straighten the beam
transverse tensile forces
the branch splits along its length along the weakest (tangential) axis, through its centre
hazard beams are particularly common where lateral roots join the trunk and wind forces can cause delamination
how trees prevent delamination?
the main longitudinal wood cells are linked radially by the rays, which are laid down plentifully where lateral stresses are high
case 2: an initially straight branch of light wood
compressive transverse stresses are set up
the light wood has very low transverse strength because of the thin-walled cell
the branch fails in compression where stresses are highest and wood weakest, along the midpoint
case 3: an initial straight branch of dense wood
transverse compressibe stregnth id big enough to stop buckling
tensile fracture vector occurs
but the crack is diverted longitudinally at the midpoint because of the low transverse tensile stregnth
greenstick fracture
same seen in childrens bones
dont know why is occures
maybe due to anisotropy of young bone, adult bone is more isotropic because of remodeling
